Negativity of Lyapunov Exponents and Convergence of Generic Random Polynomial Dynamical Systems and Random Relaxed Newton's Methods

Abstract

We investigate i.i.d. random complex dynamical systems generated by probability measures on finite unions of the loci of holomorphic families of rational maps on the Riemann sphere Ĉ. We show that under certain conditions on the families, for a generic system, (especially, for a generic random polynomial dynamical system,) for all but countably many initial values z ∈ Ĉ, for almost every sequence of maps γ = (γ1, γ2, . . .), the Lyapunov exponent of γ at z is negative. Also, we show that for a generic system, for every initial value z ∈ Ĉ, the orbit of the Dirac measure at z under the iteration of the dual map of the transition operator tends to a periodic cycle of measures in the space of probability measures on Ĉ. Note that these are new phenomena in random complex dynamics which cannot hold in deterministic complex dynamical systems. We apply the above theory and results of random complex dynamical systems to finding roots of any polynomial by random relaxed Newton's methods and we show that for any polynomial g, for any initial value z ∈ C which is not a root of g ′ , the random orbit starting with z tends to a root of g almost surely, which is the virtue of the effect of randomness.

New Q-Newton's method meets Backtracking line search: good convergence guarantee, saddle points avoidance, quadratic rate of convergence, and easy implementation

“…Here, we will compare the performance of New Q-Newton's method Backtracking against New Q-Newton's method, the usual Newton's method, BFGS, Adaptive Cubic Regularization [9,4], as well as Random damping Newton's method [13] and Inertial Newton's method [2].…”

Non-i.i.d. random holomorphic dynamical systems and the probability of tending to infinity

“…The first author introduced the random relaxed Newton's method in [17] and suggested that the random relaxed Newton method might be a more useful method to compute the roots of polynomials than the classical deterministic Newton's method. The key is that sufficiently large noise collapses bad attractors and makes the system more stable.…”

“…The phenomena are called noise-induced phenomena or randomness-induced phenomena, which are of great interest from the mathematical viewpoint. For more research on random holomorphic dynamical systems and related fields, see [2,4,7,8,10,11,14,15,17,18].…”

A fast and simple modification of Newton's method helping to avoid saddle points

“…One can choose δ n randomly at each step. To this end, we note the paper [22], where by methods in complex dynamics, it is shown that Random damping Newton's method can find all roots of a complex polynomial in 1 variable, if we choose δ n to be a complex random number so that |δ n −1| < 1, and the rate of convergence is the same as the usual Newton's method. It is hopeful that this result can be extended to systems of polynomials in higher dimensions.…”